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Definition
The bundle of unit tangent vectors of a Riemannian manifold.
Let be a Riemannian manifold and its tangent bundle. The unit tangent bundle consists of the vectors of length one:
Its projection is , .
Each fiber is the unit sphere in the inner-product space .
John M. Lee, Introduction to Riemannian Manifolds, Chapters 2–5. Publisher record.
Let be a finite-dimensional smooth manifold. A Riemannian metric on is a smooth symmetric covariant -tensor such that is a positive-definite inner product on for every . Equivalently, is a bundle metric on the tangent bundle . A Riemannian manifold is a pair . In local coordinates, , where the matrix is symmetric positive definite at every point and each coefficient is smooth.
Let be a smooth manifold of dimension .
The tangent bundle of is the set
the disjoint union of all tangent spaces of , equipped with the projection map
There is a canonical smooth manifold structure on characterized as follows: for any smooth chart on with , the induced map
sending a tangent vector to in the coordinate basis is a smooth coordinate chart onto an open subset of . These charts are compatible across a smooth atlas and make into a smooth map.